2-ADIC ARITHMETIC-GEOMETRIC MEAN AND ELLIPTIC CURVES

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1 -ADIC ARITHMETIC-GEOMETRIC MEAN AND ELLIPTIC CURVES KENSAKU KINJO, YUKEN MIYASAKA AND TAKAO YAMAZAKI 1. The arithmetic-geometric mean over R and elliptic curves We begin with a review of a relation between the arithmetic-geometric mean and an elliptic curve over R. Let a and b be positive real numbers satisfying a > b. We define two sequences {a n } and {b n } inductively by (1 a 0 := a, b 0 := b a n+1 := a n + b n, b n+1 := a n b n (n 0, where b n+1 is the positive square root of a n b n. These sequences are convergent to a common limit. We call the common limit of {a n } and {b n } the arithmetic-geometric mean of a and b, which is denoted by M(a, b. Gauss discovered amazing depth of this subject, including a connection between the arithmetic-geometric mean and elliptic integrals, which we summarize briefly. Using the sequences {a n } and {b n } introduced in (1, we define an elliptic curve E n over R for each n N by Then there exists an isogeny ( g n : E n E n+1 (x, y E n : y = x(x 1(x (a n /b n. of degree two. Hence we obtain the diagram ( (x + (a n /b n 4(a n /b n x, y(x (a n /b n 8 3 a n /b n x g 0 g 1 g 1 g n 1 g n (3 E 0 E1 E En. We employ (3 to calculate the periods of E 0. The idea is to relate them with that of a (singular curve E defined by y = x(x 1. We regard E as the limit of the family {E n } because the sequences {a n } and {b n } have the common limit. Since E is a rational curve, its period can be calculated with elementary functions. Now, let α n and β n be the real and imaginary basis of H 1 (E n (C, Z, respectively (unique up to the sign. Note that α n and β n satisfy g n α n = α n+1, g n β n = β n+1. Date: October 0,

2 Let ω n = dx/y be a basis of H 1 (E n, Ω 1 E n. The complex numbers α n ω n and β n ω n are called the periods of the elliptic curve E n. We have gnω n+1 = a n /b n ω n, and see an an 1 a0 ω n = ω 0 = b n+1 ω 0. β n b n b n 1 b 0 β 0 b 0 β 0 The right hand side of (4 converges to M(a 0 /b 0, 1 β 0 ω 0. On the other hand, the left hand side of (4 is seen to converge to the period of E β dx y = 0 dx x(x 1 = πi. Therefore we see πi ω 0 = β 0 M(a 0 /b 0, 1. For completeness sake, we record a fact that the other period can be also written by the arithmetic-geometric mean (see, for example, [3]: π ω 0 = α 0 M(1, a 0/b p-adic analogue: summary of known results and our result Henniert and Mestre [1] defined the p-adic arithmetic-geometric mean for all prime p and related it with the period of an elliptic curve over a p-adic field with multiplicative reduction. Unfortunately, their method cannot be generalized, unless p =, to elliptic curves with good reduction, essentially because the isogeny ( is of degree two. Still, it is possible to use the -adic arithmetic-geometric mean in a study of elliptic curves with good ordinary reduction over a -adic field. It is explained in [] that, given an ordinary elliptic curve Ẽ over a finite field of characteristic two, one can exploit the -adic arithmetic-geometric mean to compute the canonical lift of Ẽ under a mild technical assumption that the j-invariant j(ẽ of Ẽ does not belong to F 4. (In [], T. Satoh attributed this result to an unpublished work of R. Harley. He worked over an unramified extention K 0 of Q, because the canonical lift is defined over K 0. In this talk, we report that his method can be modified in such a way that it works over any (possibly ramified finite extension of Q. We also remove the technical assumption that j(ẽ F 4. In section 3, we introduce the arithmetic-geometric mean over a finite extention of Q. In Section 4, we study an isogeny of elliptic curves over a -adic field with good ordinary reduction. Section 5 describes the relation of the -adic arithmeticgeometric mean with the canonical lift of an ordinary elliptic curve over a finite field. 3. -adic arithmetic-geometric mean Let K be a finite extention over Q, R the valuation ring of K, π a prime of R, and v the normalized valuation of K. Put e = v(.

3 Lemma 3.1. Let ξ be a non-zero element of K satisfying v(ξ 1 e + 1. There exists a unique element η such that η = ξ and v(η 1 e + 1. We write ξ for η. We will define the -adic arithmetic-geometric mean by using Lemma 3.1. Let a and b be non-zero elements of K satisfying v((a/b 1 3e. We define the sequences {a n } and {b n } in a similar way as (1: (4 a 0 := a, b 0 := b a n+1 := a n + b n an, b n+1 := b n (n 0. b n Note that a n /b n satisfies v((a n /b n 1 3e for any n, and the definition of b n+1 makes sense by Lemma 3.1. Unlike the case of R recalled in Section 1, these sequences do not always converge. Proposition 3.. The sequences {a n } and {b n } defined in (4 converge if and only if v((a/b 1 > 3e. When this condition is satisfied, the sequences {a n } and {b n } converge to a common limit. Remark 3.3. If v((a/b 1 = 3e, then we can define the sequences {a n } and {b n }, although they do not converge. 4. Elliptic curves over -adic fields with good ordinary reduction Proposition 4.1. Let E be an elliptic curve with good ordinary reduction over K. There exist a finite extension K of K and λ K such that v (λ 1 = 3v ( and that E is isomorphic (over K to the elliptic curve (5 E λ : y = x(x 1(x λ, where v is the normalized valuation on K. Moreover, the point (0, 0 on E λ is in the kernel of the reduction map. Now we assume λ is an element of K (by replacing K with K if necessary such that v(λ 1 = 3, and let E = E λ be an elliptic curve over K with good ordinary reduction, defined by (5. The point (0, 0 is in the kernel of the reduction map. Let E be another elliptic curve define by E : y = x(x 1(x λ, λ = 1 + λ λ, where λ is defined by Lemma 3.1. We remark that λ satisfies v(λ 1 = 3e. By the same argument as (, there exists an isogeny φ from E to E such that the following diagram commutes (6 E( K red Ẽ( F φ φ E ( K red Ẽ ( F. Here Ẽ and Ẽ are the reduction of E and E respectively, K is an algebraic closure of K, and F is an algebraic closure of the residue field F of K. 3

4 Remark 4.. Let Fr : Ẽ Ẽ( be the Frobenius map. One sees that Ẽ is isomorphic to Ẽ(, and that φ is identified with Fr under this isomorphism. 5. -adic arithmetic-geometric mean and canonical lift We exploit the construction in the previous section to make a relation with the -adic arithmetic-geometric mean. Let E 0 be an elliptic curve with good ordinary reduction over K, having the Weierstrass equation E 0 : y = x(x 1(x λ 0, where λ 0 K satisfies v(λ 0 1 = 3e. We define inductively the sequence {λ n }: λ n+1 := λ n + 1 λ n (n 0 and define an elliptic curve E n for each n 0: E n : y = x(x 1(x λ n. Note that the sequence {λ n } comes from the arithmetic-geometric mean because for the sequences {a n } and {b n } defined in (4, we have a n+1 /b n+1 = (a n /b n + 1/( a n /b n for all n 0. Using the argument mentioned above, we obtain a commutative diagram: E 0 ( K φ 0 E 1 ( K φ 1 φd E d 1 ( K φ d 1 E d ( K φ d red red red red Ẽ 0 ( F Fr Ẽ( 0 ( F Fr Fr Ẽ(d 1 0 ( F Fr Ẽ(d 0 ( F = Ẽ0( F where d := [F : F ] and, for any n 0, φ n : E n E n+1 is the isogeny constructed in (6 (applied to λ = λ n. The sequence {λ n } does not always convergent, but the following theorem holds. Theorem 5.1. Let i {0, 1,, d 1}. (1 The sequence {λ dn+i } n 0 converges to an element λ i in K. ( We define an elliptic curve E i over K by the Weierstrass equation Then E i y = x(x 1(x (λ i. is the canonical lift of Ẽi. We briefly explain how ( follows from (1. We regard E i as the limit of curves {E dn+i }. One sees that E i is an elliptic curve with good ordinary reduction over K and that the reduction of E i is isomorphic to Ẽ(i 0. We obtain a commutative diagram: E0( K φ 0 E1( K φ 1 φ d E d 1 ( K φ d 1 E d ( K = E0( K red red red red red Ẽ 0 ( F Fr Ẽ( 0 ( F Fr Fr Ẽ(d 1 0 ( F Fr Ẽ0( F = Ẽ0( F where φ i is the isogeny constructed in (6 (applied to λ = λ i. Therefore φ d 1 φ 0 is a lift of the d -Frobenius endomorphism on Ẽ0 to E 0. Since the existence of a lift of the Frobenius characterizes the canonical lift, we are done. 4

5 References [1] Henniart, Guy and Mestre, Jean-François. Moyenne arithmético-géométrique p-adique, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989, no. 13, [] Satoh, Takakazu. On p-adic point counting algorithms for elliptic curves over finite fields, Algorithmic number theory (Sydney, 00, 43 66, Lecture Notes in Comput. Sci., 369, Springer, Berlin, 00. [3] Silverman, Joseph H. The arithmetic of elliptic curves. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, Mathematical Institute, Tohoku University, Aoba, Sendai , Japan address, Kensaku Kinjo: sa6m16@math.tohoku.ac.jp address, Yuken Miyasaka: sa7m7@math.tohoku.ac.jp address, Takao Yamazaki: ytakao@math.tohoku.ac.jp 5